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Mukhopadhyay A, Chakraborty S. Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games. CHAOS (WOODBURY, N.Y.) 2021; 31:023123. [PMID: 33653037 DOI: 10.1063/5.0032311] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2020] [Accepted: 01/27/2021] [Indexed: 06/12/2023]
Abstract
This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection-as a result of competition fashioned by game-theoretic consideration-into the Wright-Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker-Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.
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Affiliation(s)
- Archan Mukhopadhyay
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
| | - Sagar Chakraborty
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
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O’Sullivan JD, Knell RJ, Rossberg AG. Metacommunity‐scale biodiversity regulation and the self‐organised emergence of macroecological patterns. Ecol Lett 2019; 22:1428-1438. [DOI: 10.1111/ele.13294] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2018] [Revised: 01/21/2019] [Accepted: 05/05/2019] [Indexed: 01/07/2023]
Affiliation(s)
- Jacob D. O’Sullivan
- School of Biological and Chemical Sciences Queen Mary University of London Mile End Road LondonE1 4NS UK
| | - Robert J. Knell
- School of Biological and Chemical Sciences Queen Mary University of London Mile End Road LondonE1 4NS UK
| | - Axel G. Rossberg
- School of Biological and Chemical Sciences Queen Mary University of London Mile End Road LondonE1 4NS UK
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Galla T. Dynamically evolved community size and stability of random Lotka-Volterra ecosystems
(a). ACTA ACUST UNITED AC 2018. [DOI: 10.1209/0295-5075/123/48004] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Advani M, Bunin G, Mehta P. Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model. JOURNAL OF STATISTICAL MECHANICS (ONLINE) 2018; 2018:033406. [PMID: 30636966 PMCID: PMC6329381 DOI: 10.1088/1742-5468/aab04e] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
A central question in ecology is to understand the ecological processes that shape community structure. Niche-based theories have emphasized the important role played by competition for maintaining species diversity. Many of these insights have been derived using MacArthur's consumer resource model (MCRM) or its generalizations. Most theoretical work on the MCRM has focused on small ecosystems with a few species and resources. However theoretical insights derived from small ecosystems many not scale up large ecosystems with many resources and species because large systems with many interacting components often display new emergent behaviors that cannot be understood or deduced from analyzing smaller systems. To address these shortcomings, we develop a statistical physics inspired cavity method to analyze MCRM when both the number of species and the number of resources is large. Unlike previous work in this limit, our theory addresses resource dynamics and resource depletion and demonstrates that species generically and consistently perturb their environments and significantly modify available ecological niches. We show how our cavity approach naturally generalizes niche theory to large ecosystems by accounting for the effect of collective phenomena on species invasion and ecological stability. Our theory suggests that such phenomena are a generic feature of large, natural ecosystems and must be taken into account when analyzing and interpreting community structure. It also highlights the important role that statistical-physics inspired approaches can play in furthering our understanding of ecology.
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Affiliation(s)
| | - Guy Bunin
- Technion-Israel Institute of Technology, Haifa, 3200003, Israel
| | - Pankaj Mehta
- Dept. of Physics, Boston University, Boston, MA 02215
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Abstract
Ecological communities in heterogeneous environments assemble through the combined effect of species interaction and migration. Understanding the effect of these processes on the community properties is central to ecology. Here we study these processes for a single community subject to migration from a pool of species, with population dynamics described by the generalized Lotka-Volterra equations. We derive exact results for the phase diagram describing the dynamical behaviors, and for the diversity and species abundance distributions. A phase transition is found from a phase where a unique globally attractive fixed point exists to a phase where multiple dynamical attractors exist, leading to history-dependent community properties. The model is shown to possess a symmetry that also establishes a connection with other well-known models.
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Affiliation(s)
- Guy Bunin
- Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
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Obuchi T, Kabashima Y, Tokita K. Multiple peaks of species abundance distributions induced by sparse interactions. Phys Rev E 2016; 94:022312. [PMID: 27627322 DOI: 10.1103/physreve.94.022312] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2015] [Indexed: 06/06/2023]
Abstract
We investigate the replicator dynamics with "sparse" symmetric interactions which represent specialist-specialist interactions in ecological communities. By considering a large self-interaction u, we conduct a perturbative expansion which manifests that the nature of the interactions has a direct impact on the species abundance distribution. The central results are all species coexistence in a realistic range of the model parameters and that a certain discrete nature of the interactions induces multiple peaks in the species abundance distribution, providing the possibility of theoretically explaining multiple peaks observed in various field studies. To get more quantitative information, we also construct a non-perturbative theory which becomes exact on tree-like networks if all the species coexist, providing exact critical values of u below which extinct species emerge. Numerical simulations in various different situations are conducted and they clarify the robustness of the presented mechanism of all species coexistence and multiple peaks in the species abundance distributions.
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Affiliation(s)
- Tomoyuki Obuchi
- Department of Mathematical and Computing Science, Tokyo Institute of Technology, Yokohama 226-8502, Japan
| | - Yoshiyuki Kabashima
- Department of Mathematical and Computing Science, Tokyo Institute of Technology, Yokohama 226-8502, Japan
| | - Kei Tokita
- Graduate School of Information Science, Nagoya University, Nagoya 464-8601, Japan
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Barabás G, J Michalska-Smith M, Allesina S. The Effect of Intra- and Interspecific Competition on Coexistence in Multispecies Communities. Am Nat 2016; 188:E1-E12. [PMID: 27322128 DOI: 10.1086/686901] [Citation(s) in RCA: 91] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
For two competing species, intraspecific competition must exceed interspecific competition for coexistence. To generalize this well-known criterion to multiple competing species, one must take into account both the distribution of interaction strengths and community structure. Here we derive a multispecies generalization of the two-species rule in the context of symmetric Lotka-Volterra competition and obtain explicit stability conditions for random competitive communities. We then explore the influence of community structure on coexistence. Results show that both the most and least stabilized cases have striking global structures, with a nested pattern emerging in both cases. The distribution of intraspecific coefficients leading to the most and least stabilized communities also follows a predictable pattern that can be justified analytically. In addition, we show that the size of the parameter space allowing for feasible communities always increases with the strength of intraspecific effects in a characteristic way that is independent of the interspecific interaction structure. We conclude by discussing possible extensions of our results to nonsymmetric competition.
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9
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Analytical theory of species abundance distributions of a random community model. POPUL ECOL 2015. [DOI: 10.1007/s10144-014-0476-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Evolutionary Distributions and Competition by Way of Reaction-Diffusion and by Way of Convolution. Bull Math Biol 2013; 75:2305-23. [DOI: 10.1007/s11538-013-9890-x] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2013] [Accepted: 07/25/2013] [Indexed: 10/26/2022]
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Wu Y, Liu W, Xiao J, Zou W, Kurths J. Effects of spatial frequency distributions on amplitude death in an array of coupled Landau-Stuart oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056211. [PMID: 23004847 DOI: 10.1103/physreve.85.056211] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2012] [Indexed: 06/01/2023]
Abstract
The influences of spatial frequency distributions on complete amplitude death are explored by studying an array of diffusively coupled oscillators. We found that with all possible sets of spatial frequency distributions, the two critical coupling strengths ε(c1) (lower-bounded value) and ε(c2) (upper-bounded value) needed to get complete amplitude death exhibit a universal power law and a log-normal distribution respectively, which has long tails in both cases. This is significant for dynamics control, since large variations of ε(c1) and ε(c2) are possible for some spatial arrangements. Moreover, we explore optimal spatial distributions with the smallest (largest) ε(c1) or ε(c2).
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Affiliation(s)
- Ye Wu
- State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, China
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Noble AE, Hastings A, Fagan WF. Multivariate Moran process with Lotka-Volterra phenomenology. PHYSICAL REVIEW LETTERS 2011; 107:228101. [PMID: 22182042 DOI: 10.1103/physrevlett.107.228101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/12/2011] [Indexed: 05/31/2023]
Abstract
For a population with any given number of types, we construct a new multivariate Moran process with frequency-dependent selection and establish, analytically, a correspondence to equilibrium Lotka-Volterra phenomenology. This correspondence, on the one hand, allows us to infer the phenomenology of our Moran process based on much simpler Lokta-Volterra phenomenology and, on the other, allows us to study Lotka-Volterra dynamics within the finite populations of a Moran process. Applications to community ecology, population genetics, and evolutionary game theory are discussed.
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Affiliation(s)
- Andrew E Noble
- Department of Environmental Science and Policy, University of California, Davis, California 95616, USA
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Kelly CK, Blundell SJ, Bowler MG, Fox GA, Harvey PH, Lomas MR, Ian Woodward F. The statistical mechanics of community assembly and species distribution. THE NEW PHYTOLOGIST 2011; 191:819-827. [PMID: 21534968 DOI: 10.1111/j.1469-8137.2011.03721.x] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
• Theoretically, communities at or near their equilibrium species number resist entry of new species. Such 'biotic resistance' recently has been questioned because of successful entry of alien species into diverse natural communities. • Data on 10,409 naturalizations of 5350 plant species over 16 sites dispersed globally show exponential distributions both for species over sites and for sites over number of species shared. These exponentials signal a statistical mechanics of species distribution, assuming two conditions. First, species and sites are equivalent, either identical ('neutral') or so complex that the chance a species is in the right place at the right time is vanishingly small ('idiosyncratic'); the range of species and sites in our data disallows a neutral explanation. Secondly, the total number of naturalizations is fixed in any era by a 'regulator'. • Previous correlation of species naturalization rates with net primary productivity over time suggests that the regulator is related to productivity. • We conclude that biotic resistance is a moving ceiling, with resistance controlled by productivity. The general observation that the majority of species occur naturally at only a few sites, and only a few species occur at many sites, now has a quantitative (exponential) character, offering the study of species' distributions a previously unavailable rigor.
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Affiliation(s)
- Colleen K Kelly
- Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
| | - Stephen J Blundell
- Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
| | - Michael G Bowler
- Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
| | - Gordon A Fox
- Department of Biology, University of South Florida, 4202 E. Fowler Ave, Tampa, FL 33620, USA
| | - Paul H Harvey
- Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
| | - Mark R Lomas
- Department of Animal and Plant Sciences, University of Sheffield, Sheffield S10 2TN, UK
| | - F Ian Woodward
- Department of Animal and Plant Sciences, University of Sheffield, Sheffield S10 2TN, UK
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Yoshino Y, Galla T, Tokita K. Rank abundance relations in evolutionary dynamics of random replicators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:031924. [PMID: 18851082 DOI: 10.1103/physreve.78.031924] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2008] [Revised: 07/09/2008] [Indexed: 05/26/2023]
Abstract
We present a nonequilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multispecies replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and antisymmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93, 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher-order interactions.
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Affiliation(s)
- Yoshimi Yoshino
- Graduate School of Science and Cybermedia Center, Osaka University, Toyonaka, Osaka 560-0043, Japan.
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Pigolotti S, López C, Hernández-García E. Species clustering in competitive Lotka-Volterra models. PHYSICAL REVIEW LETTERS 2007; 98:258101. [PMID: 17678058 DOI: 10.1103/physrevlett.98.258101] [Citation(s) in RCA: 51] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2007] [Indexed: 05/16/2023]
Abstract
We study the properties of general Lotka-Volterra models with competitive interactions. The intensity of the competition depends on the position of species in an abstract niche space through an interaction kernel. We show analytically and numerically that the properties of these models change dramatically when the Fourier transform of this kernel is not positive definite, due to a pattern-forming instability. We estimate properties of the species distributions, such as the steady number of species and their spacings, for different types of interactions, including stretched exponential and constant kernels.
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Affiliation(s)
- Simone Pigolotti
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus de la Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
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